Steenrod operations

E911362

cohomology operation concept in algebraic topology

Steenrod operations are cohomology operations in algebraic topology that act on cohomology groups, providing powerful tools for distinguishing topological spaces and defining and studying characteristic classes.

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All labels observed (3)

Label	Occurrences
Steenrod squares	3
Steenrod algebra	2
Steenrod operations canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T11219603 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Steenrod operations
Context triple: [Characteristic Classes, hasSubject, Steenrod operations]

A. Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
B. Whitehead product in homotopy theory
The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
C. Stiefel–Whitney classes
Stiefel–Whitney classes are characteristic classes in algebraic topology that assign cohomology invariants to real vector bundles, capturing their topological and orientability properties.
D. Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
E. Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Steenrod operations
Target entity description: Steenrod operations are cohomology operations in algebraic topology that act on cohomology groups, providing powerful tools for distinguishing topological spaces and defining and studying characteristic classes.

A. Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
B. Whitehead product in homotopy theory
The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
C. Stiefel–Whitney classes
Stiefel–Whitney classes are characteristic classes in algebraic topology that assign cohomology invariants to real vector bundles, capturing their topological and orientability properties.
D. Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
E. Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	cohomology operation ⓘ concept in algebraic topology ⓘ
actsOn	cohomology groups ⓘ
appliesTo	cohomology of CW complexes ⓘ cohomology of topological spaces ⓘ
are	natural transformations of cohomology functors ⓘ
basisFor	structure of the mod p cohomology of spaces ⓘ
coefficientField	F_2 ⓘ F_p ⓘ
component	Bockstein homomorphism ⓘ Steenrod squares ⓘ linked to: Steenrod operations reduced pth power operations ⓘ
definedOn	cohomology with coefficients in a finite field ⓘ mod p cohomology ⓘ singular cohomology ⓘ
degreeShift	nonnegative ⓘ
field	algebraic topology ⓘ
generalizationOf	cup product powers ⓘ
introducedBy	Norman Steenrod ⓘ
invariantUnder	homotopy equivalence ⓘ
namedAfter	Norman Steenrod ⓘ
property	determine the structure of the Steenrod algebra ⓘ functorial ⓘ graded ⓘ satisfy Adem relations ⓘ satisfy Cartan formula ⓘ stable under suspension ⓘ
relatedTo	Eilenberg–MacLane spaces ⓘ Steenrod algebra ⓘ linked to: Steenrod operations cup product in cohomology ⓘ
satisfies	Adem relations among operations ⓘ Cartan formula for products ⓘ naturality with respect to continuous maps ⓘ stability with respect to suspension ⓘ
timePeriod	mid 20th century ⓘ
usedFor	defining characteristic classes ⓘ distinguishing topological spaces ⓘ studying characteristic classes ⓘ
usedIn	calculation of cohomology rings ⓘ classification of manifolds ⓘ homotopy theory ⓘ obstruction theory ⓘ stable homotopy theory ⓘ study of Postnikov towers ⓘ study of characteristic classes of vector bundles ⓘ theory of fiber bundles ⓘ
usedToDetect	nontrivial characteristic classes ⓘ nontrivial homotopy classes ⓘ

How these facts were elicited